盛揮 發問時間: 科學數學 · 7 年前

det(A)=det(A^T)數學歸納法證明

If A is a square matrix, then det(A)=der(A^T).

1 個解答

評分
  • 7 年前
    最佳解答

    A is a n*n matrix.

    proof:

    1°n=1,A=A^T

    2°Assume n=k-1, A=A^t is true.

    3°Then n=k, A is a k*k matrix.

      a11 a12 ... a1k       a11 a21 ... ak1

    A= a21 a22 ... a2k    A^t= a12 a22 ... ak2

      ... ... ... ...        ... ... ... ...

      ak1 ak2 ... akk       a1k a2k ... akk

    detA=a11detA11-a21detA21+...+(-1)^k+1ak1detAk1

    detA^t=a11det(A^t)11-a21det(A^t)21+...+(-1)^k+1ak1det(A^t)k1

    We see (A^t)ij=(Aji)^t.

    ∵Aji is a (k-1)*(k-1) matrix.

    ∴det(A^t)ij=det((Aji)^t)=detAji

    B.M.I.

    • 登入以對解答發表意見
還有問題?馬上發問,尋求解答。